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June  2012, 5(3): 641-656. doi: 10.3934/dcdss.2012.5.641

## An explicit stable numerical scheme for the $1D$ transport equation

 1 Commissariat à l’Énergie Atomique (CEA), DEN/DANS/DM2S/SFME/LETR, 91191 Gif-sur-Yvette, France

Received  August 2010 Revised  October 2010 Published  October 2011

We derive in this paper a numerical scheme in order to calculate solutions of $1D$ transport equations. This $2nd$-order scheme is based on the method of characteristics and consists of two steps: the first step is about the approximation of the foot of the characteristic curve whereas the second one deals with the computation of the solution at this point. The main idea in our scheme is to combine two $2nd$-order interpolation schemes so as to preserve the maximum principle. The resulting method is designed for classical solutions and is unconditionally stable.
Citation: Yohan Penel. An explicit stable numerical scheme for the $1D$ transport equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 641-656. doi: 10.3934/dcdss.2012.5.641
##### References:
 [1] C. Bardos, M. Bercovier and O. Pironneau, The vortex method with finite elements, Math. Comp., 36 (1981), 119-136. doi: 10.1090/S0025-5718-1981-0595046-3. [2] F. Boyer and P. Fabrie, "Éléments d'Analyse pour l'Étude de Quelques Modèles d'Écoulements de Fluides Visqueux Incompressibles,'' Mathématiques & Applications (Berlin), 52, Springer-Verlag, Berlin, 2006. [3] J. Burgers, "A Mathematical Model Illustrating the Theory of Turbulence," edited by Richard von Mises and Theodore von Kármán, Adv. Appl. Mech., Academic Press, Inc., New York, (1948), 171-199. doi: 10.1016/S0065-2156(08)70100-5. [4] S. Dellacherie, On a diphasic low Mach number system, M2AN Math. Model. Numer. Anal., 39 (2005), 487-514. doi: 10.1051/m2an:2005020. [5] B. Després and F. Lagoutière, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics, J. Sci. Comput., 16 (2001), 479-524. doi: 10.1023/A:1013298408777. [6] J. Douglas Jr. and T. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19 (1982), 871-885. [7] J. Douglas Jr., C.-S. Huang and F. Pereira, The modified method of characteristics with adjusted advection, Numer. Math., 83 (1999), 353-369. doi: 10.1007/s002110050453. [8] G. Fourestey, "Simulation Numérique et Contrôle Optimal d'Interactions Fluide Incompressible/Structure par une Méthode de Lagrange-Galerkin d'Ordre 2,'' Ph.D Thesis, École Nationale des Ponts et Chaussées, 2002. Available at: http://hal.archives-ouvertes.fr/tel-00005675. [9] E. Godlewski and P.-A. Raviart, "Numerical Approximation of Hyperbolic Systems of Conservation Laws,'' Applied Mathematical Sciences, 118, Springer-Verlag, New York, 1996. [10] F. Holly and A. Preissmann, Accurate calculation of transport in two dimensions, J. Hydr. Div., 103 (1977), 1259-1277. [11] R. LeVeque, "Numerical Methods for Conservation Laws," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. [12] J. Marsden and A. Chorin, "A Mathematical Introduction to Fluid Mechanics," Springer-Verlag, New-York-Heidelberg, 1979. [13] S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [14] Y. Penel, "Étude Théorique et Numérique de la Déformation d'une Interface Séparant deux Fluides Non-Miscibles à Bas Nombre de Mach," Ph.D Thesis, Univ. Paris 13, Available at: http://hal.archives-ouvertes.fr/tel-00547865. [15] Y. Penel, S. Dellacherie and O. Lafitte, Global solutions to the 1D Abstract Bubble Vibration model, Submitted. [16] O. Pironneau, On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. Math., 38 (1981/82), 309-332. doi: 10.1007/BF01396435.

show all references

##### References:
 [1] C. Bardos, M. Bercovier and O. Pironneau, The vortex method with finite elements, Math. Comp., 36 (1981), 119-136. doi: 10.1090/S0025-5718-1981-0595046-3. [2] F. Boyer and P. Fabrie, "Éléments d'Analyse pour l'Étude de Quelques Modèles d'Écoulements de Fluides Visqueux Incompressibles,'' Mathématiques & Applications (Berlin), 52, Springer-Verlag, Berlin, 2006. [3] J. Burgers, "A Mathematical Model Illustrating the Theory of Turbulence," edited by Richard von Mises and Theodore von Kármán, Adv. Appl. Mech., Academic Press, Inc., New York, (1948), 171-199. doi: 10.1016/S0065-2156(08)70100-5. [4] S. Dellacherie, On a diphasic low Mach number system, M2AN Math. Model. Numer. Anal., 39 (2005), 487-514. doi: 10.1051/m2an:2005020. [5] B. Després and F. Lagoutière, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics, J. Sci. Comput., 16 (2001), 479-524. doi: 10.1023/A:1013298408777. [6] J. Douglas Jr. and T. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19 (1982), 871-885. [7] J. Douglas Jr., C.-S. Huang and F. Pereira, The modified method of characteristics with adjusted advection, Numer. Math., 83 (1999), 353-369. doi: 10.1007/s002110050453. [8] G. Fourestey, "Simulation Numérique et Contrôle Optimal d'Interactions Fluide Incompressible/Structure par une Méthode de Lagrange-Galerkin d'Ordre 2,'' Ph.D Thesis, École Nationale des Ponts et Chaussées, 2002. Available at: http://hal.archives-ouvertes.fr/tel-00005675. [9] E. Godlewski and P.-A. Raviart, "Numerical Approximation of Hyperbolic Systems of Conservation Laws,'' Applied Mathematical Sciences, 118, Springer-Verlag, New York, 1996. [10] F. Holly and A. Preissmann, Accurate calculation of transport in two dimensions, J. Hydr. Div., 103 (1977), 1259-1277. [11] R. LeVeque, "Numerical Methods for Conservation Laws," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. [12] J. Marsden and A. Chorin, "A Mathematical Introduction to Fluid Mechanics," Springer-Verlag, New-York-Heidelberg, 1979. [13] S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [14] Y. Penel, "Étude Théorique et Numérique de la Déformation d'une Interface Séparant deux Fluides Non-Miscibles à Bas Nombre de Mach," Ph.D Thesis, Univ. Paris 13, Available at: http://hal.archives-ouvertes.fr/tel-00547865. [15] Y. Penel, S. Dellacherie and O. Lafitte, Global solutions to the 1D Abstract Bubble Vibration model, Submitted. [16] O. Pironneau, On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. Math., 38 (1981/82), 309-332. doi: 10.1007/BF01396435.
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